scalars and vectors - School of Physics - The University … vectors are equal if they have the same...

HSC PHYSICS ONLINE 1

HSC PHYSICS ONLINE

THE LANGUAGE OF PHYSICS:

SCALARS and VECTORS

SCALAR QUANTITIES

Physical quantities that require only a number and a unit for their

complete specification are known as scalar quantities.

mass of Pat mPat = 75.2 kg

Pat’s temperature TPat = 37.4 oC

Pat’s height hPat = 1555 mm

Symbols and subscript are a convenient way to identify the physical

quantity (symbol) and object (subscript).

Fig.1. Scalar temperature field. At each location, the

temperature is specified by a number in oC.

http://www.physics.usyd.edu.au/teach_res/hsp/sp/spHome.htm



Fig. 2. Scalar rainfall field. At each location, the rainfall is

specified by a number in mm.

In physics, a scalar field is a region in space such that each point in

the space a number can be assigned. Examples of scalar fields are

shown in figure (1) and (2) for temperature and rainfall distributions

in Australia respectively.



VECTOR QUANTITIES

Physical quantities that require for their complete specification a

positive scalar quantity (magnitude) and a direction are called

vector quantities.

Today the wind at Sydney airport is

-1 o

35 km.h 33 N of Ev

Fig.2. A magnitude and direction is need to specify the wind.

The black lines represent the pressure (scalar) and the red

arrows the wind (vector). The length of an arrows is proportional

to the magnitude of the wind and the direction of the arrow gives

the wind direction.



A vector quantity can be visualized as a straight arrow.

The length of the arrow being proportional to the

magnitude and the direction of the arrow gives the

direction of the vector.

A vector quantity is written as a bold symbol or a

small arrow above the symbol. Often a curved line

draw under the symbol is used when the vector is

hand written.

When dealing with vectors it is a good idea to define a frame of

reference to specify the vector and its components. Figure (3) shows

the [2D] vector s and its components ,x y

s s .

Fig.3. Frame of reference used to specify the vector s and its

components ,x y

s s .



VECTOR ALGEBRA

Fig.4. Vector algebra.

The magnitude of a vector is a positive scalar. The magnitude

of the vector A is written as A or A .

In [2D] the A vector can be expressed in terms of its

components ,x y

A A and unit vectors ˆ ˆ,i j

2 2

ˆ ˆ cos sin

tan

x y x y

y

x y

x

A A i A j A A A A

AA A A A

A

Two vectors are equal if they have the same magnitude and

small direction A B .

The negative of any vector is a vector of the same magnitude

and opposite in direction. The vectors A and A are

antiparallel.

Multiplication of a vector by a scalar A . The new vector has

the same direction and a magnitude A .



Vector addition: vectors can be added using a scaled diagram

where the vectors are added in a tail-to-head method or by

adding the components

ˆ ˆx x y y

R P Q P Q i P Q j

Vector subrtaction: can be found by using the rule of vector

addition

ˆ ˆx x y y

S P Q P Q P Q i P Q j



Two vectors can’t be multiplied together like two scalar

quantities. Only vectors of the same physical type can be added

or subtracted. But vectors of different types can be combined

through scalar multiplication (dot product) and vector

multiplication (cross product).

Scalar product or dot product of the vectors A and B is

defined as

cosC A B AB

The projection or component of A on the line containing B is

cosA . The angle between the two vectors is always a

positive quantity and is always less than or equal to 180o. Thus,

the scalar product can be either positive, negative or zero,

depending on the angle between the two vectors

o0 180 1 cos 1 .

The result of the scalar product is a scalar quantity. If two

vectors are perpendicular to each other, then the scalar product

is zero cos(90 ) 0o . This is a wonderful test to see if two

vectors are perpendicular to each other. If the two vectors are

in the same direction, then the scalar product is A B

cos(0 ) 1o .



The vector product or cross product of two vectors A and B is

defined as

ˆsinC A B AB n

The magnitude of the vector C is sinC C A B .

The vector n̂ is a unit vector which is perpendicular to both the

vectors A and B .

The angle between the two vectors is always less than or equal

to 180o. The sine over this range of angles is never negative,

hence the magnitude of the vector product is always positive or

zero o0 180 0 sin 1 .

The direction of the vector product is perpendicular to both the

vectors A and B . The direction is given by the right hand

screw rule. The thumb of the right hand gives the direction of

the vector product as the fingers of the right hand rotate from

along the direction of the vector A towards the direction of the

vector B .



VECTOR EQUATIONS

Consider the motion of an object moving in a plane with a uniform

acceleration in the time interval t. The physical quantities describing

the motion are

Time interval t [s]

Displacement s [m]

Initial velocity u [m.s-1]

Final velocity v [m.s-1]

Acceleration a [m.s-2]

The equation describing the velocity as a function of time involves the

vector addition of two vectors

x x x y y y

v u a t v u a t v u a t

The equation describing the displacement as a function of time

involves the vector addition of two vectors

2 2 21 1 1

2 2 2x x x y y ys u t a t s u t a t s u t a t

The velocity as a function of displacement.

Warning: the equation stated in the syllabus is totally incorrect

2 2

2v u a s this equation is absolute nonsense

Two vectors can’t be multiplied together. The correct equation has to

show the scalar product between two vectors

2v v u u a s

This equation should not be given in vector form but expressed as

two separate equations, one for the X components and one for the Y

components

2 2 2 2

2 2x x x x y y y y

v u a s v u a s



Work and the scalar product

Consider a tractor pulling a

crate across a surface as

shown in figure (5).

Fig. 5. A crate being pulled by a tractor.

We want to setup a simple model to consider the energy transferred

to the crate by the tractor. In physics, to model a physical situation,

one introduces a number of simplifications and approximations. So

we will assume that the crate is pulled along a frictionless surface by

a constant force acting along the rope joining the tractor and crate.

We then draw an annotated scientific diagram of the situation

showing our frame of reference. The crate becomes the system for

our investigation and the system is drawn as a dot and the forces

acting on the system are given by arrows as shown in figure (6).

Fig. 6. The system is the cart (brown dot). The forces acting on

the system are the force of gravity G

F , the normal force N

F and

the tension of the rope T

F .



Energy is transferred to the system by the action of the forces doing

work on the system. Work is often said to be equal to a force

multiplied by a distance W Fd . This is a poor definition of work. A

much better definition of the work done by a constant force causing

an object to move along a straight line is to use the idea of scalar

(dot) product

cosW F s F s work is a scalar quantity

where the angle is the angle between the two vectors The angle

between the two vectors is always a positive quantity and is always

less than or equal to 180o hence 1 1 .

Work done by the gravitational force and by the normal force are zero

because the angle between the force vectors and displacement vector

is 90o (cos90o = 0).

The work done by the tension force is

cos cosT T T Tx

W F s F s F s F s

and so the work done on the system is the component of the force

parallel to the displacement vector multiplied by the magnitude of the

displacement.

The concept of the scalar product is not often used at the high school

level, but, by being familiar with the concept of the scalar product

you will have a much better understanding of the physics associated

with motion.



Torque and the vector product

What is the physics of opening a door?

It is the torque applied to the door that is

important and not the force.

A force can cause an object to move and a torque can cause an

object to rotate. A torque is often thought of as a force multiplied by

a distance. However, using the idea of the vector (cross) product we

can precisely define what we mean by the concept of torque.

ˆsinr F r F n

The vector is the torque applied, the vector r is the lever arm

distance from the pivot point to the point of application of the force

F . The angle is the angle between the vectors r and F . The

direction of the torque n̂ is found by applying the right hand screw

rule: the thumb points in the direction of the torque as you rotate

the fingers of the right hand from along the line of the vector r to the

vector F . The torque is perpendicular to both the position vector r

and the force F .



The concepts of unit vectors, scalar product and vector product are

not covered in the syllabus. However, having a more in-depth

knowledge will help you in having a better understanding of physics

and will lead to a better performance in your HSC examination.


scalars and vectors - School of Physics - The University … vectors are equal if they have the same...

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